{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

LectureFeb29_MATH4321_12S

LectureFeb29_MATH4321_12S - Recall that Equilibrium...

Info icon This preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
Recall that Equilibrium Principle: BR to each other Maximin Principle: Safety first For Player I: Find p so that Min q p T Aq is largest. p is called the Safety Strategy or Optimal Strategy. Min q p T Aq is the lower value. For Player II: Find q so that Max p p T Aq is smallest. q is called the Safety Strategy or Optimal Strategy. Max p p T Aq is the upper value. Minimax Theorem: Maximin=Minimax
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Theorem : (Maximin Principle Equilibrium Principle) Let p ~ belong to X *, q ~ belong to Y *, be safety strategies for Player I and II respectively. Then, p ~ , q ~ are best responses to each other i.e. <p ~ , q ~> is an equilibrium pair.
Image of page 2
Proof: p ~ is a safety strategy for Player I means Min q p ~T Aq = Max p Min q p T Aq q ~ is a safety strategy for Player II means Max p p T Aq ~ = Min q Max p p T Aq
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Then, we show in the following p ~ is a BR to q ~ . p ~T Aq ~ Min q p ~T Aq = Max p Min q p T Aq = Min q Max p p T Aq = Max p p T Aq ~ p T Aq ~ , for any p in X * This completes the proof. Similarly, q ~ is a BR to p ~ .
Image of page 4
Theorem (Equilibrium Principle Maximin Principle): Let p ~ , q ~ be best responses to each other i.e. <p ~ , q ~ > is an equilibrium pair. Then, p ~ (in X * ), q ~ (in Y * )are safety strategies for Player I and II respectively.
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Proof: As p ~ is a BR to q ~ , we have p ~T Aq ~ = Max p p T Aq ~ Max p Min q p T Aq = Min q Max p p T Aq Min q p ~T Aq = p ~T Aq ~ (q ~ is a BR to p ~ ) Thus, Min q p ~T Aq = Max p Min q p T Aq and p ~ is a safety strategy for Player I. Similarly, q ~ is a safety strategy for Player II.
Image of page 6
Image of page 7

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Theorem: Let <p 1 , q 1 >, <p 2 , q 2 > be equilibrium pairs. Then, <p 1 , q 2 > is also an equilibrium pair.
Image of page 8
Image of page 9
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern